for every morphism $f:A \to B$ in $\mathbf{C}$.
We shall often write $\alpha:F\to G: \mathbf{C}\to \mathbf{D}$ to indicate that $\alpha$ is a natural transformation between two functors $\mathbf{C}\to \mathbf{D}$.
The composite of two natural transformations $\alpha:F\to G$ and $\beta:G\to H$ in the category $[\mathbf{C},\mathbf{D}]$ is the natural transformation $\beta \alpha:F\to H$ obtained by putting $(\beta\alpha)_A=\beta_{A}\alpha_A$ for every object $A\in \mathbf{C}$,
The category $[\mathbf{C},\mathbf{D}]$ is locally small if $\mathbf{C}$ is small and $\mathbf{D}$ locally small, in which case we shall often denote it by $\mathbf{D}^{\mathbf{C}}$. The category $\mathbf{D}^{\mathbf{C}}$ is small, if additionally $\mathbf{D}$ is small.
Recall that a presheaf? on a small category $C$ is defined to be a contravariant functor? $C\to \mathbf{Set}$, or equivalently a covariant functor $C^o\to \mathbf{Set}$. If $X$ and $Y$ are presheaves on $C$, then a map $X\to Y$ is defined to be a natural transformation $X\to Y$. The category of presheaves $[C^o,\mathbf{Set}]$ is locally small and we shall denote it by $\mathbf{P}(C)$.
Recall that a simplicial set is defined to be a presheaf on the simplicial category $\Delta$. We shall denote the category of simplicial sets $\mathbf{P}(\Delta^o)$ by $\mathbf{SSet}$.
If $K\mathbf{Vect}$ is the category of $K$-vector spaces over a field $K$ and $G$ is a group, then the functor category $[\mathbf{B}G,K\mathbf{Vect}]$ is the category of a $K$-linear representations of $G$.
Let $I$ be the category defined by the poset $[1]=\{0,1\}$, and let $i$ be the unique arrow $0\to 1$. If $\mathbf{C}$ is a category, then a functor $X:I\to \mathbf{C}$ the same thing as a morphism $x:X_0\to X_1$ in $\mathbf{C}$, where $X_0=X 0$, $X_1=X 1$ and $x=X(i)$. If $Y:I\to \mathbf{C}$ is another morphism $y:Y_0\to Y_1$, then a natural transformation $f:X\to Y$ is a commutative square
in the category $\mathbf{C}$. The category $[I,\mathbf{C}]= \mathbf{C}^I$ is called the arrow category of $\mathbf{C}$.